From a Google search I think your laptop has the ESS ES1878 chipset, which is similar to the ES1879 chipset in my Sony PCG-505TS laptop (Pentium1-300).
These are on the ISA bus, and are not auto-detected.

First reset all bios settings to defaults. While in bios, also change "PnP OS" to "NO" if possible.
Now run the ALSA configuration wizard (alsaconf) and select the legacy "es18xx" driver (which is actually the ALSA snd-es18xx module).

If still no sound, you may require the special workaround that I needed, to force the snd-es18xx driver to accept the correct DMA values. I described this workaround in June 2006 here -
http://www.murga-linux.com/puppy/viewtopic.php?t=8816

Those resource values should be listed in your bios, but the following values seem to be quite common -
irq=5 dma1=1 dma2=5

And don't forget to add the "isapnp=0" parameter to your options line. This will stop the snd-es18xx driver from auto-detecting your resource values, which is important since this function is known to fail with early ESS chipsets.

Are you using gxine? I think that gxine will be far too "heavy" for a 150MHz CPU.
Even XMMS may be too big.
I recommend the commandline mp3 audio player, madplay, which Puppy already contains.
And you can install a simple gui for madplay called "xhippo" from here -
http://puppyfiles.org/dotpupsde/dotpups/Multimedia/xhippo-3.3-gtk2.pup

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